Algebraic Groups : Arithmetic and Geometry Friday, March ... · Special session \Algebraic Groups : Arithmetic and Geometry" Friday, March 13th, 2020. Let X be a smooth projective

On the integral Tate conjecture for 1-cycles on theproduct of a curve and a surface over a finite field

Joint work with Federico Scavia (UBC, Vancouver)

Jean-Louis Colliot-Thelene(CNRS et Universite Paris-Sud ⊂ Paris-Saclay)

AMS sectional meeting, CharlottesvilleSpecial session “Algebraic Groups : Arithmetic and Geometry”

Friday, March 13th, 2020

Let X be a smooth projective variety over a finite field F of char. p.Let ` be a prime, ` 6= p. For any i ≥ 0, there is a cycle map

CH i (X )⊗ Z` → H2i (X ,Z`(i))

from the Chow groups of codimension i cycles to the projectivelimit of the etale cohomology groups H2i (X , µ⊗i`n ), which is aZ`-module of finite type.For i = 1, this map reads

Pic(X )⊗ Z` → H2(X ,Z`(1))

and Tate conjectured that it is always surjective. This is related tothe conjectured finiteness of Tate-Shafarevich groups of abelianvarieties over a global field.It is known for geometrically separably unirational varieties (easy),for abelian varieties (Tate) and for most K3-surfaces.

For i > 1, Tate conjectured that the cycle map

CH i (X )⊗Q` → H2i (X ,Q`(i))

is surjective. Very little is known. One may give examples wherethe statement with integral coefficients does not hold. However,for X of dimension d , it is unknown whether the Integral Tateconjecture for 1-cycles, henceforth denoted T1 = T d−1

(T1) The cycle map CHd−1(X )⊗ Z` → H2d−2(X ,Z`(d − 1)) isonto.

may fail.

For d = 2, this is a special case of the Tate conjecture.

For arbitrary d , the integral Tate conjecture for 1-cycles holds forX of any dimension d ≥ 3 if it holds for any X of dimension 3.

For X of dimension 3, some nontrivial cases have been established.• X is a conic bundle over a geometrically ruled surface (Parimalaand Suresh).• X is the product of a curve of arbitrary genus and ageometrically rational surface (Pirutka).

For smooth projective varieties X over C, there is a formallyparallel conjecture for cycle maps

CH i (X )→ Hdg2i (X ,Z)

where Hdg2i (X ,Z) ⊂ H2iBetti (X ,Z) is the subgroup of rationally

Hodge classes. The conjecture with Q-coefficients is a famousopen problem. With integral coefficients, several counterexampleswere given, even with dim(X ) = 3 and 1-cycles.A recent counterexample involves the product X = E × S of anelliptic curve E and an Enriques surface. For fixed S , provided E is“very general”, then the integral Hodge conjecture fails for X(Benoist-Ottem). The proof uses the fact that the torsion of thePicard group of such a surface is nontrivial, it is Z/2.

It is reasonable to investigate the Tate conjecture for cycles ofcodimension i ≥ 2 assuming T 1 : The conjecture is true for cyclesof codimension 1 over any smooth projective variety.

Theorem (CT-Scavia 2020). Let F be a finite field, F a Galoisclosure, G = Gal(F/F). Let E/F be an elliptic curve and S/F bean Enriques surface. Let X = E ×F S . Let ` be a prime differentfrom p = char.(F). Assume T 1.If ` 6= 2, or if ` = 2 but E (F) has no nontrivial 2-torsion, then themap CH2(X )⊗ Z` → H4(X ,Z`(2)) is onto.

We actually prove a general theorem, for the product X = C × Sof a curve C and a surface S which is geometrically CH0-trivial,which here means :Over any algebraically closed field Ω of F, the degree mapCH0(SΩ)→ Z is an isomorphism.In that case Pic(SΩ) is a finitely generated abelian group.

For F a Galois closure of F, G = Gal(F/F), and J the jacobian ofC , still assuming T 1, we prove that CH2(X )⊗ Z` → H4(X ,Z`(2))is onto under the condition HomG (Pic(SF)tors, J(F)) = 0.

We do not know whether this condition is necessary.

In the rest of the talk, I shall sketch some ingredients of the proof.

Let M be a finite Galois-module over a field k . Given a smooth,projective, integral variety X/k with function field k(X ), and i ≥ 1an integer, one lets

H inr (k(X ),M) := Ker[H i (k(X ),M)→ ⊕x∈X (1)H i−1(k(x),M(−1))]

Here k(x) is the residue field at a codimension 1 point x ∈ X , thecohomology is Galois cohomology of field, and the maps on theright hand side are “residue maps”.

One is interested in M = µ⊗j`n , for which M(−1) = µ⊗(j−1)`n and in

the direct limit Q`/Z`(j) = limjµ⊗j`n , for which the cohomology

groups are the limit of the cohomology groups.

Theorem (Kahn, CT-Kahn) For X/F smooth, projective ofarbitrary dimension, the torsion subroup of the (conjecturallyfinite) group

Coker[CH2(X )⊗ Z` → H4(X ,Z`(2))]

is isomorphic to the quotient of H3nr(F(X ),Q`/Z`(2)) by its

maximal divisible subgroup.

A basic exact sequence (Kahn, CT-Kahn). Let F be an algebraicclosure of F, let X = X ×F F and G = Gal(F/F).

For X/F a smooth, projective, geometrically connected variety overa finite field, long exact sequence

0→ Ker[CH2(X )` → CH2(X )`]→ H1(F,H2(X ,Q`/Z`(2)))

→ Ker[H3nr(F(X ),Q`/Z`(2))→ H3

nr(F(X ),Q`/Z`(2))]

→ Coker[CH2(X )→ CH2(X )G ]` → 0.

The proof relies on work of Bloch and on the Merkurjev-Suslintheorem. Via Deligne’s theorem on the Weil conjectures, one has

H1(F,H2(X ,Q`/Z`(2))) = H1(F,H3(X ,Z`(2))tors)

and this is finite.

For X a curve, all groups are zero.For X a surface, H3(F(X ),Q`/Z`(2)) = 0.For X/F a surface, one also has

H3nr(F(X ),Q`/Z`(2)) = 0.

This vanishing was remarked in the early stages of higher class fieldtheory (CT-Sansuc-Soule, K. Kato, in the 80s). It uses a theoremof S. Lang, which relies on Tchebotarev’s theorem.

For our 3-folds X = C × S , S as above, we have an isomorphismof finite groups

Coker[CH2(X )⊗ Z` → H4(X ,Z`(2))] ' H3nr(F(X ),Q`/Z`(2)),

and, under the assumption T 1 for all surfaces over a finite field, atheorem of Chad Schoen implies H3

nr(F(X ),Q`/Z`(2)) = 0.

Under T 1 for all surfaces, for our threefolds X = C × S with Sgeometrically CH0-trivial, we thus have an exact sequence of finitegroups

0→ Ker[CH2(X )` → CH2(X )`]→ H1(F,H2(X ,Q`/Z`(2)))

θX−→ H3nr(F(X ),Q`/Z`(2))→ Coker[CH2(X )→ CH2(X )G ]` → 0.

Under these hypotheses, the surjectivity of

CH2(X )⊗ Z` → H4(X ,Z`(2))

(integral Tate conjecture) is therefore equivalent to thecombination of two hypotheses :

Hypothesis 1The composite map

ρX : H1(F,H2(X ,Q`/Z`(2)))→ H3(F(X ),Q`/Z`(2))

of θX and H3nr(F(X ),Q`/Z`(2)) ⊂ H3(F(X ),Q`/Z`(2)) vanishes.

Hypothesis 2 Coker[CH2(X )→ CH2(X )G ]` = 0.

(Optional slide)

Hypothesis 1 is equivalent to :

Hypothesis 1a. The (injective) map from

Ker[CH2(X )` → CH2(X )`]

to the (finite) group

H1(F,H2(X ,Q`/Z`(2))) ' H1(F,H3(X ,Z`(2))tors)

is onto.

In dimension > 2 , we do not see how to establish the validity ofthis hypothesis directly – unless of course when the finite groupH3(X ,Z`(2))tors vanishes. The group H3(X ,Z`(1))tors is thenondivisible part of the `-primary Brauer group of X .

Discussion of Hypothesis 1 :The map ρX : H1(F,H2(X ,Q`/Z`(2)))→ H3(F(X ),Q`/Z`(2))vanishes.This map is the composite of the Hochschild-Serre map

H1(F,H2(X ,Q`/Z`(2)))→ H3(X ,Q`/Z`(2)))

with the restriction to the generic point of X .

We prove :Theorem. Let Y and Z be two smooth, projective geometricallyconnected varieties over a finite field F. Let X = Y ×F Z . Assumethat the Neron-Severi group of Z is free with trivial Galois action.If the maps ρY and ρZ vanish, then so does the map ρX .

Corollary. For the product X of a surface and arbitrary manycurves, the map ρX vanishes.

On must study H1(F,H2(X , µ⊗2`n )) under restriction from X to its

generic point.As may be expected, the proof uses a Kunneth formula, along withstandard properties of Galois cohomology of a finite field.As a matter of fact, it is an unusual Kunneth formula, withcoefficients Z/`n, n > 1. That it holds for H2 of the product oftwo smooth, projective varieties over an algebraically closed field, isa recent result of Skorobogatov and Zarhin, who used it in another context (the Brauer-Manin set of a product).

Discussion of Hypothesis 2 :

Coker[CH2(X )→ CH2(X )G ]` = 0.Here we restrict to the special situation : C is a curve and S isgeometrically CH0-trivial surface.One lets K = F(C ) and L = F(C ).On considers the projection X = C × S → C , with generic fibrethe K -surface SK . Restriction to the generic fibre gives a naturalmap from

Coker[CH2(X )→ CH2(X )G ]`

toCoker[CH2(SK )→ CH2(SL)G ]`.

Using the hypothesis that S is geometrically CH0-trivial, whichimplies b1 = 0 and b2 − ρ = 0 (Betti number bi , rank ρ ofNeron-Severi group), one proves :

Theorem. The natural, exact localisation sequence

Pic(C )⊗ Pic(S)→ CH2(X )→ CH2(SL)→ 0.

may be extended on the left with a finite p-group.

(Optional slide)

To prove this, we use correspondences on the product C × S , overF.We use various pull-back maps, push-forward maps, intersectionmaps of cycle classes :

Pic(C )⊗ Pic(S)→ Pic(X )⊗ Pic(X )→ CH2(X )

CH2(X )⊗Pic(S)→ CH2(X )⊗Pic(X )→ CH3(X ) = CH0(X )→ CH0(C )

Pic(C )⊗ Pic(S)→ CH2(X ) = CH1(X )→ CH1(S) = Pic(S)

Not completely standard properties of G -lattices for G = Gal(F/F)applied to the (up to p-torsion) exact sequence of G -modules

0→ Pic(C )⊗ Pic(S)→ CH2(X )→ CH2(SL)→ 0

then lead to :

Theorem. The natural map from Coker[CH2(X )→ CH2(X )G ]`to Coker[CH2(SK )→ CH2(SL)G ]` is an isomorphism.

One is thus left with controlling this group. Under theCH0-triviality hypothesis for S , it coincides with

Coker[CH2(SK )` → CH2(SL)`G ].

At this point, for a geometrically CH0-trivial surface overL = F(C ), which is a field of cohomological dimension 1, using theK -theoretic mechanism, one may produce an exact sequenceparallel to the basic exact sequence over F which we saw at thebeginning. In the particular case of the constant surfaceSL = S ×F L, the left hand side of this sequence gives an injection

0→ A0(SL)` → H1(L,H3(S ,Z`(2)`)

where A0(SL) ⊂ CH2(SL) is the subgroup of classes of zero-cyclesof degree zero on the L-surface SL.

Study of this situation over completions of F(C ) (Raskind) and agood reduction argument in the weak Mordell-Weil style, plus afurther identification of torsion groups in cohomology of surfacesover an algebraically closed field then yield a Galois embedding

A0(SL)` → HomZ(Pic(S)`, J(C )(F)),

hence an embedding

A0(SL)`G → HomG (Pic(S)`, J(C )(F)).

If this group HomG (Pic(S)`, J(C )(F)) vanishes, then

Coker[CH2(SK )` → CH2(SL)`G ] = 0

henceCoker[CH2(X )→ CH2(X )G ]` = 0,

which is Hypothesis 2, and completes the proof of the theorem :

Theorem (CT/Scavia) Let F be a finite field, G = Gal(F/F). Let `be a prime, ` 6= char.(F). Let C be a smooth projective curve overF, let J/F be its jacobian, and let S/F be a smooth, projective,geometrically CH0-trivial surface.Assume the usual Tate conjecture for codimension 1 cycles onvarieties over a finite field.Under the assumption

HomG (Pic(SF)`, J(F)) = 0,

the cycle class map CH2(X )⊗ Z` → H4et(X ,Z`(2)) is onto.